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<b>let </b><font color="Maroon">s1</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">s2</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">t1</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">t2</font> be   <a href="real_1.html#NM1">Real</a>;<br/>







<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><i><font color="Green">E28</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">qc</font> where <font color="Olive">qc</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : (  not <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">qc</font> <a href="euclid.html#K21">`1</a>  or  not <font color="Olive">qc</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> or  not <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">qc</font> <a href="euclid.html#K22">`2</a>  or  not <font color="Olive">qc</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">q</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:38_1_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; (  not <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  or  not <font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> or  not <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  or  not <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> ) )
 ;<br/><a NAME="E4:38_1_1"/>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; (  not <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  or  not <font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> or  not <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  or  not <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> ) )
 <b>by </b><i>, <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/><b>hence </b><a NAME="E5:38_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s2a</font>,<font color="Olive">t2a</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s2a</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">t2a</font> is   <a href="real_1.html#NM1">Real</a> : (  not <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s2a</font> or  not <font color="Olive">s2a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> or  not <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t2a</font> or  not <font color="Olive">t2a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> ) </span>}</span> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">sb</font>,<font color="Olive">tb</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">sb</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">tb</font> is   <a href="real_1.html#NM1">Real</a> : (  not <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">sb</font> or  not <font color="Olive">sb</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> or  not <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">tb</font> or  not <font color="Olive">tb</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">sb</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">tb</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:38_1_2"/><i><font color="Green">E31</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">sb</font>,<font color="Maroon">tb</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; (  not <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">sb</font> or  not <font color="Maroon">sb</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> or  not <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">tb</font> or  not <font color="Maroon">tb</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> ) )
 ;<br/><b>set </b><font color="Maroon">qa</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">sb</font>,<font color="Maroon">tb</font></span><a href="euclid.html#K23">]|</a></span>;<br/><a NAME="E4:38_1_2"/>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">sb</font>,<font color="Maroon">tb</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; (  not <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">sb</font>,<font color="Maroon">tb</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  or  not <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">sb</font>,<font color="Maroon">tb</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> or  not <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">sb</font>,<font color="Maroon">tb</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  or  not <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">sb</font>,<font color="Maroon">tb</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> ) )
 <b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E5:38_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">qc</font> where <font color="Olive">qc</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : (  not <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">qc</font> <a href="euclid.html#K21">`1</a>  or  not <font color="Olive">qc</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> or  not <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">qc</font> <a href="euclid.html#K22">`2</a>  or  not <font color="Olive">qc</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> ) </span>}</span> 
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:38_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">qc</font> where <font color="Olive">qc</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : (  not <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">qc</font> <a href="euclid.html#K21">`1</a>  or  not <font color="Olive">qc</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> or  not <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">qc</font> <a href="euclid.html#K22">`2</a>  or  not <font color="Olive">qc</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> ) </span>}</span>  iff <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">sb</font>,<font color="Olive">tb</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">sb</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">tb</font> is   <a href="real_1.html#NM1">Real</a> : (  not <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">sb</font> or  not <font color="Olive">sb</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> or  not <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">tb</font> or  not <font color="Olive">tb</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> ) </span>}</span>  )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:38"/>
<span class="p1">{<span class="default"> <font color="Olive">qc</font> where <font color="Olive">qc</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : (  not <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">qc</font> <a href="euclid.html#K21">`1</a>  or  not <font color="Olive">qc</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> or  not <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">qc</font> <a href="euclid.html#K22">`2</a>  or  not <font color="Olive">qc</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">sb</font>,<font color="Olive">tb</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">sb</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">tb</font> is   <a href="real_1.html#NM1">Real</a> : (  not <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">sb</font> or  not <font color="Olive">sb</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> or  not <font color="Maroon">t1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">tb</font> or  not <font color="Olive">tb</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t2</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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